Theorem prnz 4739

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4724	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4299	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2145   ≠ wne 2960  Vcvv 3457  ∅c0 4288  {cpr 4587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-un 3912  df-nul 4289  df-sn 4586  df-pr 4588

This theorem is referenced by:  opnz  5446  propssopi  5482  fiint  9274  wilthlem2  27191  upgrbi  29352  wlkvtxiedg  29883  shincli  31623  chincli  31721  constrextdg2lem  34055  spr0nelg  48080  sprvalpwn0  48087